We present a method to compute the Fermi function of the Hamiltonian for a system of independent fermions based on an exact decomposition of the grand-canonical potential. This scheme does not rely on the localization of the orbitals and is insensitive to ill-conditioned Hamiltonians. It lends itself naturally to linear scaling as soon as the sparsity of the system's density matrix is exploited. By using a combination of polynomial expansion and Newton-like iterative techniques, an arbitrarily large number of terms can be employed in the expansion, overcoming some of the difficulties encountered in previous papers. Moreover, this hybrid approach allows us to obtain a very favorable scaling of the computational cost with increasing inverse temperature, which makes the method competitive with other Fermi operator expansion techniques. After performing an in-depth theoretical analysis of computational cost and accuracy, we test our approach on the density functional theory Hamiltonian for the metallic phase of the LiAl alloy.